The University of Southampton
University of Southampton Institutional Repository

Reductive group schemes, the Greenberg functor, and associated algebraic groups

Reductive group schemes, the Greenberg functor, and associated algebraic groups
Reductive group schemes, the Greenberg functor, and associated algebraic groups
Let A be an Artinian local ring with algebraically closed residue field k, and let G be an affine smooth group scheme over A. The Greenberg functor F associates to G a linear algebraic group G := (FG)(k) over k, such that G = G(A). We prove that if G is a reductive group scheme over A, and T is a maximal torus of G, then T is a Cartan subgroup of G, and every Cartan subgroup of G is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for smooth affine schemes with reduced fibre over A, and that the Greenberg functor preserves certain normaliser group schemes over A. Moreover, we prove that if G is reductive and P is a parabolic subgroup of G, then P is a self-normalising subgroup of G, and if B and B0 are two Borel subgroups of G, then the corresponding subgroups B and B0 are conjugate in G
1-14
Stasinski, Alexander
94bd8be7-4b4f-4e22-875b-3628d8c2ca19
Stasinski, Alexander
94bd8be7-4b4f-4e22-875b-3628d8c2ca19

Stasinski, Alexander (2010) Reductive group schemes, the Greenberg functor, and associated algebraic groups. Preprint, 1-14.

Record type: Article

Abstract

Let A be an Artinian local ring with algebraically closed residue field k, and let G be an affine smooth group scheme over A. The Greenberg functor F associates to G a linear algebraic group G := (FG)(k) over k, such that G = G(A). We prove that if G is a reductive group scheme over A, and T is a maximal torus of G, then T is a Cartan subgroup of G, and every Cartan subgroup of G is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for smooth affine schemes with reduced fibre over A, and that the Greenberg functor preserves certain normaliser group schemes over A. Moreover, we prove that if G is reductive and P is a parabolic subgroup of G, then P is a self-normalising subgroup of G, and if B and B0 are two Borel subgroups of G, then the corresponding subgroups B and B0 are conjugate in G

This record has no associated files available for download.

More information

Published date: 18 March 2010

Identifiers

Local EPrints ID: 79816
URI: http://eprints.soton.ac.uk/id/eprint/79816
PURE UUID: a85b8191-4bb1-41f4-b2c4-4e9d7ec96274

Catalogue record

Date deposited: 22 Mar 2010
Last modified: 10 Dec 2021 17:35

Export record

Contributors

Author: Alexander Stasinski

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×